Chap. 7] 



GRAVITATIONAL METHODS 



215 



(kdm)/r, the gradients and curvature values, or the second derivatives of 

 the potential, are 



xz' 





dm- 



dm 



yz' 



> (7-66) 



Assume that the center of gravity of the balance beam is at a distance f 

 and the mass element at an elevation h above ground. Then by substitu- 

 tion of r = Vp^ + (r - ^Y and z' = r - /i 



V^ 



U. 



= Zkf 



dm 



a;(r - h) 



= 3fc / dm .— 



[p2 + (f _ hy^m 



yit - h) 



2 + (f - hy] 



6/2 



U: 



U, 



= 3kf 

 = 3kf 



dm 



dm 



With polar coordinates x = p cos a and y = p sin a 



U. 



= 3fc/ 

 = 3fc/ 



dm 



dm 



[p' + (r - hy] 



6/2 



-U^ = 3k f 

 2U,y = 3k j 



dm 



dm 



Fig. 7-75. Sectorial mass element. 



For a sectorial mass element (Fig. 7-75), dppdadh-8 (8 = density) 

 bounded by two concentric circles with the radii Pn and pa+i and angles 

 «„ and am+i, 



= S / / / dapdpdh. (7-67) 



•'am Jon •'0 



dm 



